Spectral bounds for the torsion function
M. van den Berg · arXiv (Cornell University) · 2017
Let $Ω$ be an open set in Euclidean space $\R^m,\, m=2,3,...$, and let $v_Ω$ denote the torsion function for $Ω$. It is known that $v_Ω$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $\Leb^2(Ω)$, denoted by $λ(Ω)$, is bounded away from $0$. It is shown that the previously obtained bound $\|v_Ω\|_{\Leb^{\infty}(Ω)}λ(Ω)\ge 1$ is sharp: for $m\in\{2,3,...\}$, and any $ε>0$ we construct an open, bounded and connected set $Ω_ε\subset \R^m$ such that $\|v_{Ω_ε}\|_{\Leb^{\infty}(Ω_ε)} λ(Ω_ε)<1+ε$. An upper bound for $v_Ω$ is obtained for planar, convex sets in Euclidean space $M=\R^2$, which is sharp in the limit of elongation. For a complete, non-compact, $m$-dimensional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary it is shown that $v_Ω$ is bounded if and only if the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator acting in $\Leb^2(Ω)$ is bounded away from $0$.